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Full text of "THEORETICAL SUBSONIC PERFORMANCE OF A DUCTED ROCKET"

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N63:86365 



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JET PROPULSION LABORATORY 

California Institute OF Technology 
Pasadena California 



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'USm OF GO^TMSS 

I, StMBDary , , ,.-... 1 

II, lafcyodtocfeloa, . . , , 2 

III, H©tati©E. ........................... 7 

1?, 0¥®r-AU Ferforisaaee of a dieted Roek®t ,...*... ..,,10 

?. Perfoyimno® Caleulatioas, 14 

A. GalcUlatiea of the Preperties ®f Air Entering tixe 
Coaibustioa Chiamter ia fei^a ®f free Stream Osadltioas ... 15 

B. Charaoterl sties of the Worklag fluid LsaTiag ttis 
Oeabustiea Chamber ia Sera® ®f Iadl?id'ual Properties 

of the laterlag Air aad Eeclcet Exhaiuit Gases. ....... 19 

1» fhe Combustion Chamber, .....,.,,.. 19 

2. Matheoatioal fdraTxLation of the problem 22 

3, tte locket Motor. . 28 

k,y luamrical ?alue ©f H , the Heat added by Oo^letelj 

Burning the MckBt Ixhaust ia Air ..,..<, 30 

G. Calculatioa of Properties of the Working Jluld al the 

End of the. la® jet l02 ale |S@iM#a 4f iit-ftriTs «f^^^-0S&:,mt 

'■ ■a®>.ao-ftl;-@-lntraac9,.<See*l0a*3^ •*':.% -. ^■'«.-^..*«* * . . ._* * 53 

¥1, fhrust Galeulatioaa , » ... 35 

aeferaaoes. ...'.,.,.,.. , k3 

figores , . . , kS 





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MStP or f ISIIHIS 
Figure . ?age 

1, Gy®s8 Seetiea Oirou^ Bact©4 Hocfeet .,,... H6 



2. ia::^p»ii|att«a Factor as a FoBctioa ef M2 aa* *it® Paraa^ter 

) . Mq » for 1® Secondary Burning ®f tiis B©ek©t felmust. G©ia- 

i 'b'QsUon (loW Mixing) Chamber Dit^asiana Are Yaria"bl0» Per* 

Bitting tte latiy q£ Only So Much Air as W®uld B® Squired 
t0,S©ali2Q Complete S&eond&rj Btaaiag of the locket Mtmxel 
1 Qasss at Ivery fli^t feloclty, kl 



3. Augwntatl on factor as a lanetion of ^^ and the ParaB»ter 
] Mo • Ooatostion Ohaaber Diaensloas Ar® farlablo, Permits 
1 tli^ths Itttiy of jttsi Ba^tt^'Air t® ««aiis5© G&^Ib%@. Second- 
ary Burning of the Hocteet Exhaust Oases at lyery fXlght 
J Veloslty. .^. . * " ,■ . . ■ . 4S 



k, A'Qgfflsntation factor a® a fimetlon of M-j, and the Parameters 

Mo an4 A f showing the Increase in Thrust ndiea More Air than 

ttat Ie©a®d for Complete Secondary Comhustion Enters the 

Dttct. , . it9 

5. Augmentation Factor as a function of rA^and the Paiaaeter A 

for the Constant V&Ivb Mj* O.iO 50 

6. Augmentation Factor a® a function of A and the Parameter M© 
for the Constant ?alu©. Mz* 0.1 ......'... 5I 

7. Axigmeatatioa Factor as a Foactloa of Mj and the Parwketer A 

for ^e Constant faltae N\o » 1. ............... 52 

S,9* AtJgB^ntatloa factor as a function of Mq and Certain 53s 

10,11. Selected Amounts of Heat Added to the Working fluid, 5I, 

for fz/fo ' 1.5, 2. 3. ^ BespeetiTely. ........... 55, 5& 

12. Compartson of Augmentation factors, Ducted locfet a»d 

Bamjet of Equal Physical Dimensions and fuel C«»0uqptlon, 

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fills report Is coaeeraed vith an ©adsaTor to aredac® to c&lculatloa th® 
; optlDiB® perforfflaac© of a jet pro|mlsion aystsm consisting ©f & coa?®ntionaX 

'') rookdt motor inserted in a raajet-typ® duct *» a so-called dueted rocUst - 

when in subaoaic flt#t at sea ISTel, 
I Obaermtioaa of liquid propellent rocket motors reveal that the ezhaust 

^see litjrn for eoii^ dietaace dowastreaa of the exhaust aozgle, and it is 
) anticipated that a similar secondary comhustioa will take place in the &%m>s** 

1 pheric air flowing throw^ the doct. 

fhe restriction on the fligfet speed of the j©t->-dnct oomMaat ion allows 
1 the prohlem of thrust computations to be conveaiently separated into three 

parts I 
i (A) tte air flow through the diffuser 

<B) fhe mixing and burning processes inside the combustion chsraber 
Co) The dlschar^ of the irarking fluid throu^ a suitable aosale 
i into the atmosphere. 

fhe final results of the performance ealctjiatioas are presented as curiree 
giving the Paugrontation factor" (the ratio of thrusts for tt^ rocket-ramjet 
coAblaatlon and the rocket motor alone) as a function of different independent 
■ • mriables. 

Of prim® interest is the static tharust of the duo te brocket power plant, 
a® preliminary perforiwace study of this special case (idiloh, in an eapaaded 
J, form, will be Vte subject of a later seaport) indicates ttet the propeXlaat 

gases will induce and accelerate a flow of air throu^ the duet, in this 



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s»im0r producing a ti^rust. A prsparator^ apprdximate calcxilatlon sbo%rs tMt 
taadar static conditioBs the tJarust of a dusted rocket, without a nozgle^ will 
t>© ^tveea 60 per c®nt and 75 par o®at of ttiat of th® norisal rocket, lajactiag 
and 'burning additional gasoline in tiaa coml>ustion chaail>er will cause &@ 
static thrust of ths diausted rocket to increase asymptotieally, to th® full rated 
parforiaanca of th# rocket motor aloas. 

Sfti© psrforpincs of the rocket— ramjet comhinatlon in f li^t is encouragla^. 
Among th® examples ehos@n to illustrate applications of the general performance 
equations is that of a clucted rocket herein the total ener^ of the rocket 
exhaust gases is released as a consequence of their comhustion vith the atmos- 
pheric oxi^gen flowing throi^ the Smat. fhrust coiaputations for this case 
re-yeal that a suttahle ramjet-type duct surrounding a liqUiA-propellant rocket 
may h® instrxuMntal in causing tSM thrust of the noraal rocket to he aaltiplied 
almost fourfold. Compared to a ramjet of equal diffuser are® ratio, equal 
eomhustion chamher area, and equal consumption of fuel, the ducted rocket ^n 
a greater thrust over the whole suhsonie range of flight T^locities. 

The ramjet, or athodyd (aero-thermodynamic-duct), consists of a diffuser 
in «iiich atmospheric air is compressed, a combustion chamher within iiiiich 4 
suitable fuel la injected into th® air and burned, and a. nossl® through which 
the products of the subsequent reaction are expelled. Tk@ simplest thermojet 
siachine, after the rocket, is the ramjet, jsiac^ it requires no device for 
drawing air throu^ the. duct eats^nce, the air being forced into the diffuser 
by tlte i^iMilne action resulting from the forward motion of tii® unit. 







II 

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P« P, IM. Project ICC^-S? lilK-^ WtH^jHyil ^S® 3 

OBDCIf Project 

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On th© basis of a theor®tieal perforimao® aaalysia, it is anticipated 
that a large field of application to miBsilea and transonic and sapersoais 
aircraft awaits tha dsvelopaent of the raajet powr plant (Of ^fs 1 and 2). 

I®v®rth8l®8B, sine© th® co^rasaion r«sttltii% from the rawning of air is 
tho chi@f factor in presenting the products of coritoiftion from disctorgiag 
tlirough th® forward (diffuaer) opening, forcing ti^m to be exhausted by vmj 
of the no%zl&, it is apparent that at sero airspeed the ramjet vith an open 
I diffuaer entrance can produce no useful fori«(ard thrust. 

Hence it is necessary to solve the problem of accelerating a ramjet off 

1 ' ■' ■ . 

I the ground and increasing the velocity to that at which combustion will pro- 

ceed satisfactorily and a net forward Idfltrust will be maintained. The take- 

■> \ off and initial stage of a ramjet-powered fll^t could be executed in aa 

i obvious way by the use of an auxiliary power plant, such as externally fitted 

rocket boosters, as in the no^ familiar JATO installations. 

• However, E, S. fsien of the Jet Propulsion laboratory, OMiOIf , sug^sted 

tlmt a rocket motor discharging into the forward part of the ramjet combustion 

i 

chamber could not only stupply the thr\mt required to accelerate the ramjet 

* up to a velocity at %«hich compression by ram wuld becoH® effective, but also 

mi#it act as an ejector jet, inducing a flov of air which would perhaps even 

t augment the thrust of the normal rocket. 

f 

Such a coabiaation of a conventional rocket motor and a ramjet #ict (con* 
i sieting of diffuser, combustion ehasA>er, and dischar^ nozzle) ^111 be termed 



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ORDCIT Project i. !-, ' 

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fh® elemeataiy th®ory of ths throast augmsator la l»s®d osi tli© follewii^ 
Icteatlties, which relate the kinetic ®n®r^ of a mass m moving /with a Telocity 
V to ths momentum of the mme mas si 

Therefore, correspondiag to a given kinetic energy^ the fflomentum is relatiTely 



I great when the "vis -riva" is diffused throti^ a lar^ ma.BB of fluid moving 



bomparatively slowly, and relatively small when the ener^ is concentrated in 
a slight mass possessing a high velocity. 

The ordinary augmentor, which is t>as®d on the principle ahov®, seeks 
to enhance the thrust of a given jet hy diffusing the kinetic energy of th® 
exhaust Into a larger mass of fluid, the Work required for the acceleration 
of the sacontery flow being drawn from ths otherwise wasted kinetic ener^ 
of the Jet, which would trail away behind without contributing to th® thrust. 

for the conventional type of thrust augment or , analytical and ezperi" 
mental studies indicate that only the thrust of a static Jet, •'when th® induced 
air stream reaches the ejector at about atmospheric pressure and with negli- 
gible v®loclty®8 can be considerably augmented (Cf fief 3). Further, "when 
the approach pressure or velocity of the air increases, the au^aentatioa of 
thrust falls off rapidly; and it appears fr®ra these tests that, with a Jet 
velocity of 1000 ft per sec, an approach velocity of lOO mph is sufficient 
to reduce th® gain to a negligible amount. At higher approach velocities the 
effect of the ejectpr is practically nil, there beixjg no appreciable gain 
or 1©88«« (Cf lef 3). 

The results quoted above were obtained with augmentors employing con- 
verging guides ahead of the mixing section; the ducted rocket, lAlch has a 







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diverging seetion in front of the mixing chancers thus differs from th@ aag- 
mentor in an ingjortant respect. Heac®, it canaot 'he anticipst@d that at low 
speeds the dtscted rocket thrust idll he superior to tJoat of the free jet| In 
fact, at sfflall fll^t velocitie® the ijaflTieac® of tim duct surrounding th® 
rocket is such as to causa the thrust of th® latter to decrease. The amount 
hy which the thrust of the aonoal rocket is redaced at lo¥ speeds is an im- 
portant pBohlSffi .which will h® dealt with in a 0uhs#quent report. 

Besides offering a solution to th®. problem of launching a ramjet-powered 
aircraft, a rocket motor peraanently installed within a duct possesses certain 
attractive features of its own, lot the least of these is the possihility of 
using the flaming gases wnich issue from liquid propellant rocket® as the 
means of starting comhust ion and of maintaining continuous ramjet operation. 

Theoretical and experimental studies of rocket motors utilizing liquid 
fuels r@T®al tibat the exit ^aes contain large qtantitles of hydro^n Mid 
carhon monoxide (in addition to incomhustihiles such as nitrogen, iiater vapor, 
and carbon diozid® [^Cf fief k] ). 5!he heats of combustion of these ^ses ar® 
high I for example, the exhaust of a rocket motor designed for a mixture of 
liquid oxygen-gasoline, in the ratio '}t2 "b^ weight, at a chamber pressure of 
300 psia, yields 5bOO Btu per lb of primary reactants when completely "burned, 
fhe long jet of flame observed issuing from liquid propellant rockets is th® 
result of the reaction of the H,. and CO with o;^g®n in the air. 

In practice this ener^ is wasted, so a device (such as a duct surroisnd- 
ing the jet) which could develop added thrust efficiently from the heat re- 
leased by the secondary coifibuation ©f th® propellant ^ses with air would b« 
an improveaent over existing rocket installation® j if the increased d^ag du® 



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1 to th® dttot did aot sender th® sebes^ iBrpractlc&l. 

' V Moreover, a@ subsequent calculatloas will show, at hlgi^r fli^l speeds 

mora air eaters th® ramjet cooibustion chamber than Is required for the cobs- 

i - 

plete (secoBdary) buraing of ttie rock;®* exhaust gases. !Qi®A, to utilize tte 
i exeese inflowing air to better adYantage, it sight prove depirable to Inject 

additional fuel, sucli as ^soline, into Idie combustion dJbiamber. 

from the foregoing, it is clear that eaerg^^ can be added to the air 
flowing .throu^ the duet in either one or both of two wayss 

(1) By burning the combustible rocket exhaust gases in the air 
12) By burning additional fuel in the air. 
laeentially, the two processes differ onl^r in the respect that the jet ^ses 
in (1), entering tne combustion chamber at a velocity greater than that at 
which the fuel in (2) wouM be injected, possess more momentum than the added 
fuel. But even the iao!iientiu& of the exbaust gases in (1} is saall compared 
ts the mm of momenta and presstjre forces acting on the primary and secondary 
fluids in the combustion chamber, the ratio of the two having the order of 1 
to SO. &i!ic@, in the first approximation both means of adding ener^ to the 
dnct. air are considered to be equivalent » and the eacact manasr in %Siich ths 
heat is introduced is not important. 

In any event, a performance analysis of the ducted rocket seemed warrant- 
ed, and this report presents the first results of such a study. 

It is reasonable to assume (as the optimum , condition under liiich only 
the rocket exhaust is burned) that the propellaat ^ses are completely eoa- 
suMid in the presence of the atmoepheric osygen flowing through the dnct and 
to eoB^put® the performance of the ducted rocket on this premise. For the most 



a'AAii 




I ■ ^ 



1 

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part, 1^® «5aloxilatioBs to foil®? are p3«dl©al®d oa tiiis ^sg^^tloa. 

Tb^ middis p©rtio& of the dm;t will 1)@ desi^mted th® eoslmstioB ebaaber, 

siHGd it is priiB&rilF %s®d as t}i@ sectios «^@r@iii ^@ p»)p@ll^at ^ess &r@ 

Imrned. 

farther, tii© roctot @3^iaust gases will ts called the primry fltildi tt® 

inflowing air will "b® referred to as the seeonasry fluid? aad th® aistwr« #f 

tialritni0d air aad prod^ts of the seeondary reaction will %@ called %h® work- 

iag fluid. 

fh®re imn no a priori iadicatioa that th© dtycted rosket could eoa^jet® 
I with o1^®r jet propulsion schemes, so all calculatioas wer® directed toimrd 

SKalimtlag the upper limit to the perforwace. of th® rocket-ramjet ^mt e©»- 
■ ^Inatioa. H®bc®, all theraodyuamic procsse®® are coasidj^red perfect, frl6ti®a- 

j loss flow of ideal compreasiM® fluids is assumed throu^out, hB&% less®® ar® 

Qsglectfd, aad chemical reactions are postulated to, h® carried to completioa* 
\ All computations are mad® for suhsoai^c free fli^t velocities at sea 

level altitude, th® latter restrietioa heing arhitrarj aad serving only t© 
j particularize th® results. 



fhe notation adhered to in this report is saamariEsd t}@low. 
A Batio of th® i^ss of air allowed to eater th® doct to the aass of 

air required for complete s®coadary combustioa of the rocket exhaust 

gases, ' 
c Iffeetlve exhauttt velocity of the rocket i®t, ft per see 
Cp Specific heat of gas at constant pressure, Btu per lb °H 



ynB^if^ 



•1 









, Cv Specif ie heat ©f ^s at eoastsat Toltlai, B%a p®r lb \ 

D l^ag force 

E Total eaargy of ^s, per 11> s»s« 

f Area, sq f t 

f^ 6ro®B seetienal «r®a of toroat of tte rocket aozzle, sq ft 

') F fhraat, lb 

J A©c©l®r&tioa of graylty, 32,2 ft psr a©©*^ 

] H Heat added to tlie wrkiag fluid- la the eoBt^stioa ekambsr, owr 

5 , ■ ■ 



aad above that preasat ia the rocket exhaust, Bta 
m Mass of fluid flowiag past a givea plea© ia uait ti»@, slug® p@r eu ft 
M ifeeh aumberj tti© ratio of the fluid Telocity to the local 

velocity of souad 
p Absolut© pressure, lb per sq ft 

r *latio of the mass of air eateriag the coabustioa chamber to tte 
vm,BB of burat rocket gates ejected thro^igh the rocket aostlt, 
all ia uait time 
R tes coastaat, Btu par slug % 
T Absolut© teaperature, ®B 
V felooity, ft psr sec 
t lati© of specific heats of a ga»» Cp/Cv 
17 Iffieieaey factor 
§ Iteaaity, slugs per cu ft, 

©xe subseriptB c, o, i, z, a, 4 have the folltidag si^lflCBBceai 
( )c will deaote coaditloa© ia ^© rocket combustioa ehaaber 
( )q will cteaote ooadltioas ia the staadard ati^si^ere, far froa the 






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, 4lstT»^iag effect of tti® <teiet 

^ , C ), »111 a®siot® ooadilioa® at th@ ®atmac® t© tJa® difffeser (Of Fig X) 

' ^ ^2 ^^^ d@a®te ceaditlQELS at th© end of the diffuses, aestissd to 'b® 

in tb® mxm plane as t|i0 '@xit of ^® rocket mo2zl@ and th® eatraae® 
^ ■ ■ ■ 

^ t© th© oom^ustioa diambsr 

,, { )^ 1^111 disaote coadilioas at thie ooaclusioa of the com^ustlea cimmb&Tt 

' asstused to "b® also tae aatraaace to the dact aoaala 

I ( L will deaote coaditioas at the exit of t^e Sxtjst aozzl®, 

fh® superscripts ()»(),() %^111 sigttify tti® followiagi 

( ) will dsaot© ooaditioas refsrriag to air j th© saeoadary flxild 

( )' 'Will dsaot® coadltions referriag to the gases la the rocket j®tg 

%h.@ primary fluid' . 

( ) vlll deaot® coadltions referriag to th@ workiag fluid, 

for axa^le, 

/, equal to 1.4, is th® (coastaat) ratio ®f specific h®ats for air 

y\ equal to 1.2S5 (Of a©f 3), ia the a&sus^d coastaat ratio of 

specific heats of th® rocket ®:^must gases 

• y"j the ratio of specific heats for th® worklag fluid, Is asstaasd to 

■fe® a msaa Talus, gtv@a by 

Ia eoaformity with th® aotatioa, m^ aad m^' ar® th® msn&n of air aad 
pjpspallsat pi8@s, r®8p©oti-?®ly, ©a taring ^@ ramjet ooaMstloa chaajher ia 
uait tim®. 



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fSie Id«al4ae4 eroi» $©cUoa of what sd^t be a typleal duct@d?-roek®t 
p®«®r plaat is slxewa in fig 1, 

If it is &ssti®®d timt all of tks mw^teuaion takes plaes in tlia diffuser 
(rasarsly t© ae^iST® eis^lieity ©f id®as and langoagu), air with a imleclty Vq 
created lay th@ motion of t]bis aircraft anters thi^ raajst dmt at tlia prsssv® 
Po of ^© ataosphora. fh® difftiasr is ©oasid®r®d to comprsss the air 
adiatetieallj, the final -mines of the Telocity and pressure "being y^ and 
p^ , rei^eetiTely. 

After ®3!h&XLB%ing into the com'bastion cham1)er, the rocket ^ses are 
'bomed in tibie presence of air aspirated ^u'oo^ the duct entrance, the enero" 
released going partly to ele-mte the teaperattire of the working fluid, aa,d 
partly to accelerate that mixture, 

Becat^e of tiie dieeii^tion of ener^ in tnrMlsnt misii^, friction, and 
the acceleration of the working fluid, there will h® a certain drop in &e 
pressure of the aiacture, as it lasses throng the comlfeUBtion chaiaher, fron 
the mlu® p^ to p^ . 

fh@ nozzle then expands the gases f ro4 th® pressure p^' at tbs end of 
the comibustion chamher to timt of the atmo sphere, po > After XesTlng %e 
dnct, the axhaust g^ses eool down to the temperature of the atmoei&ere, waet« 
i!^ their heat ener^. fhe thrust augpentor ftaiotioas t© tranefer efficiently 
as mush as possi'ble of ^e aonAntuta and ener^ of the jet to a greater aaas 
©f air, in an endea-ror to increase the ae^entxaa. Since the wm.e@ of m,tter 
leaving the jet is si&all compared with the m^BB of air flowing throt;^ the 



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, ^iy 



Pag© 11 



tebst» the Inereass in moaentiim Is prie^rlly d®p@na^iit ^on the aaoqut lir 
irMdti the -relecity at &e s^rajdt nozzle exit @3£e@©da th@ fli^t Tsloolty of 
1^ d^et. Uadsr suite.))!® eeaditloas ths eslmust TSlcMslly V4" will t)® ^eats? 
thao. th© fr@9 fli^t speed Vo;a thrust, or force in the direettaa of motioa 
of the duct, i@ iixQ direct z^eult of the positlTe chaage ia aomeatua, ae idll 
h© ahowa helow. ^ 

lather thaa calculate the thrust of a a»»t®4 rocket (or of aaj reactloa 
proimlsilF® system) as the MEOuat "b^ ^hieh the moffleatum of the working fluid 
leairlag the aozzle is greater thaa the BOMatum of air eateriag the diffueerj 
it is mor® accurate to compute the ^rust "bj a coaaideratioa of the forces 
astiag oa the fluid paasiag through and ©T©r the duct, 

la the sketch helow the stream tube is takea so far in front of and aimj 
from th® duet that the disturl»ing effect of the latter on the preesure hae 
vanished, fhere is circular syimnetry at every plaa® seetioa normal t® the 
axis, and outside ©f the tuhe free stream coadltione obtain, 





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fli® net ttougt of th© dtieted roci^t - tii® actuai ^rwt loss th® dmg - 
Is ttiea th® exs@08 of prassur© force aad Mss^atutt of the flMd mt ieotlon Ij., 
which eoateiss feh@ aozzls exit, ovfia? the pressure forces aad axial momeata of 
the fl^d at the lateral strean tulna surface aad Sectloa 0, ^ere ^e outside 
air dlffmsioa cosaneacea. 

§iace th® quaatity ©f mass lea-ring the duet is greater thaa that ^M.ch 
eaters I hy aa amouat eqtial to the mass of propell@at sst^ ejected from the 
rocket, the sum of areas f4 * fi, will he greater than the total &x@& fo + fj. , 

3f deflaltioa, ao fluid passes; through the lateral surface of the tuhe, 
eo that ao momeatum is traaeferred aormal to the Tmll* Eo^@ver, hecause ths' 
generators of the tube surface are aot parallel to the axis, there will he 
a corapoaeat of pressure actiag ia the directioa parallel to the axis, furttor, 
siace the pressures at Sectioas aad k are the same, p^ » the pressure force 
oa Sectioa k es-ceede that oa Section hy aa amouat givea hy the prodsjct of 
Po aad the differeace ia the areas at the two seetioas, 

fh® magaitude of t^e differeace ia areas at Sectioas aad k is pre«> 
cisel^ the projectioa of the lateral surface of %h& stream tuhe oa Seotioa k^ 
wad ttierefore tiis pressure force oTer the area hy which -f^ + ^(, exceeds fo + *s 
is equal aad opposite to the pressure force oa the streaa tuhe's lateral 
8«pfae«. 

Upea application of the gsaeral thrust foraula, which states tJmt the 
net thrust is f^ differeace hetiieea the pressure force aad morasatvm at 
Sectloa If aad ^e aaiial coiiQ>oaeats (deaoted hy a secoad su'bscript a ) of ^tm 
pressus^ forces aad moaeata takea orer th© lateral streaa tuhe surface ( ^ (. ) 
aad Seetioa 0, it is clear that the 



;- -i- 



p. p. tM, Probst' 'ilgaf 




■ '■ m^^v% *I6'V 5^-3 



Pfti@ 13 



^■f4 -'HFfe -^o ^r^ ^'f^ 

a p®8ltiY8 ^lto«st btiag a fore® T»giag tto -aait ta a dlrsetias opposite to 
tliat of tb@ oaln flair. 

Siaoe tl3,® Intsgral of a mm i@ the mm of iatogmls, l^b^e xmXatlon a^o^r® 

Ne-fc thrust = P^dfi-h^v^if i- h>i,A^ * P^K'^df - Ift^^ -AoVe**^* -/ Pf*** 

fli* pr®ss«ras oTsr fell® areas f © » ^4 » ^y » ^t. aa^e tise samo ®ad squal to p^ , so 

fp^df +fri<if -fi%df -fps^f = /Pod-f i 

^ ^U \ \ "^^s '^f.+V^o-^j 

aeeordlBg to one of the consldsrationa a'bo'?^, this forc« is eqiml a&4 0H>#sit@ 
t® J Pj-a*^^ » *^ axial coatpoaeBt of the pressBre fors© tipon the lateral 
surfacd of th@ free stream tube, and heaoe there reiaaine 

Net thrust = J9^^^ df +J9i,yl'i^ -J^X^^ 'J^s'^s<i^ • 

*4 ^b ^o ^S 

Her® the coadition f9C^i,\i'^^-^> wMeh aeans that there Is ao aziial aomaat«m 
fliax throii^ ^ ^alls, has been iarposed. 

Z& the usual definition the thrust of a reaction propulsi^re sjceteffl is 
giTsa as &e exeese of mooentuai leaviag Idie ejrstem orer '^lat wb^ioh eaters. 
With this iaterprttatioii the formula for the thruet of ths ducted roc^i^t 
eensiets of ^e teras 

■|^,. | ,. ■■■„ ■^-.■■■ , .^.^— ,. ........ — j.^..- , f.. ^ .^j^ nrr r iiii fi i i i r' 'T i ' ii 1 1 " ii T-ii iiii n iii ir'- i i " ii 'n i i « ii T fWiTT«iiiiii.wwiwnrnw«fc«iiwHM)P»rii n i n »»w«iiiiiiiw «ii ■! i - -i-i - i i i. ■■ i i i ii r i ., i ,. i j- l innj iiii i m i iiiiiii lWwij i ..tLuiu iinjjiii»iii|ii]Liniiiiiiiii.iMii| n i«i 

•fhe aacial coa^ioaent of momentum f lust through the latest stS'faee of the 
etreaa tube is represented as ^^^t'^'-'^*-**'^' being ttoe iategrai.* over i&m 
lateral aiorfao®, of the mass flow ttoot^h ths surface multiplied by the 
axial eoffiponeat of tb« velocity. 








' . ■' '' , - ^ ' M >; "■'. i 



or the doaage lia mome'ntiJHJ of the fluid passing throxi^ the d\ict» Tha qiaantity 
/ K'^k'^^-J ^s^s^'^^ } raprssantiag th® asmeat^ffl afean^ aacpwrieseed W *Ji® 

fluid ia leasing over th© test, is ssseatially the exUraaJ drag fore* (ia th® 
a.\}stt]»ee of a pr®8«tstre differeac® %•(«•«& Sactioae 6 and 5) <t^^ ^U &0%>s 9X» 
lilc^ili^ <mlbtaat9d» aine» the ma^itmdo will depend upon tli« size and sonf ig^ 
«yatioB of a particular iastallatioa aad its relatioarfilp t© the oth®r eo^oaaat 
parts of a projectile or aircraft. 

fhea the force ©xerted ©a tha fluid pass lag throu^ the diet, heretefor® 
called the thrust, is givea "^ . 

It Aould "be aoted that this quaatity is only dsfiaed as the thrust aad does 
aot represeat the aet forc^ imparted to the duet Tjy the air affected as a 
result of the motioa of the tact throu^ the atmosphere, 

©1® theoretical perforaaac® aaalysis of the ducted rocket is ©oayeaieatly 
divided iato three parts t 

Ca) froperties of the air aateriag the eonibustioa chaaljer are 
ottaiaed ia terms of free stream coaditioas. 
(':l CB) Oharacteristlco of the working fluid Isaviag the oofflMstion 
ehaaher are calculated f% ftoctioas ef properlleis ®f the 
primary aad secoadary fluids eateriag the middle seetioa. * 
(C) *Sk& aozzle is dssigaed t® reduce t^e pressure of the ws*lag 
fluid to that of the ataso^her®. 








!Sias® separat® parts of the lnY@stlgatioa as« 3P@port©4 t3g>0n l)®l©w, 

from the idealised orasa asction iDf the docted rocket pois®r piaat, lig 1, 
it is @Tlds&t that the large @ad ©f th® dlffu@@r aad th@ entrai»3® to %lm G@m^ 
hUBtioB chaia'ber ars as suae d to ^ m» and th@ saos seetioia; a lils:@ statesMtat 
is true of the combustion chamber @xit and t^e noEzle entraa@e. 

Sine© "th® force exerted V » fluid ©a a body tdeponds] . . . only ea th® 
relatiT© velocity hetHsen tii©®® (Cf Sef 5)j it follows ^lat th® ducted rock®t 
Hiay h» ohosea at i»m%t with the air ffloTiag oTsf and throu^ th@ docl. 

At an iafinit© di«tanos from the duct» and at the saiM height abo7® th@ 
siirfae® of the earth ~ here ooasldAilid as plane - the felooity @f th@ air is 
assumed to he uniform and parallel to the axis of the rocket. Atmos|^@ri@ 
conditions are taten as standard, based on tiae HA6A Ataospheric Tables 
(f.S, lo, 21g» I.A.C.A. , 1925)1 furttiermore, onlj subsoate fli^t Toloclties 
are considered. 

In all performance calculations sea leirel atw>s|^eric properties will ' 
be used J for which 

Acceleration of gravity = q = 32.2 ft i»r s®e^ 

Density =%= O.OO237S slugs per eu ft 

Pressure = po= 21l6 lb per sq f t 

Batio of specific heats for air =y= l*¥ 

feaperature =To= 51S,l| "aankiae 

Velocity of sound =yj^-^ = 112© ft per se©, 

A. Calculation of the Properttes of Air Baterlng fee Qoabustion Chiwaber 
in f ermg of l^ee Stream (^ofidition^ 

fh® oalculatioa of the properties of air entering t]^ coabustlon ohaBber 





p. p. UB. 'Project «527 > UMW^Ib i J^P ' '^ ■ ?«S@ 1^ 

(ffiSeif Project 

JPIr-Q4L0If 



irtie«ld 'b© i^parated iato two parts $ first, the fr®s %%tm.m diffuBloa o^ght 
to "ba dlsctissed, and seeond, properties of the air at the diffuser exit should 
he given as fonetions of those at the dlffuser entrance. 

I© render peesihle the analysis of the free stream diffusioa, given 
properties of adr at the diffuser entrance in terms of free stream ceniitions, 
the folXetdng idealizations are ©mploTedi 

(1) fhe air is postulated to tie a perfect gas with constant 
specific heats 

(2) All thenaod^fnamlo processes are considered to he adiahatic 
vith the exponent 1.4 

(3) She free stream diffusion is assumed 100 per cent efficient, 
farther, only suhsonic fli^t velocities of the dacte(i rocket are considered | 
hence no shocks can occur in front of the diffuser. 

If the additional requirement is imposed that tlM -^slcal processes are 
restricted to those, which conserve miss, a ^t of slntQltaaeous. equations to 

T, f, 

l3e solved for M, and ^^ in terns of the parameters M^ and — , are ©htaiasd. 

However, the analysis of expanding fluid flow in a duct is more coii^lte, 
hecause even a well designed diffuser will have Inevitahle |>»ea«BPe losses, 
the presence of vhich tend to i^ke the transforioation of kinetic ener^ isto 
pressure energy less efficient than it would he where there is no de^adatioa 
of ener^. In this way the concept of diffuser efficiesffly, origiEslly so^sst- 
ed hy Stodola, is introaaeed, heing defined as 

representing the fraction of kinetic ener^ chan^ useful for ptirposes ©f 
coaspregslon (referred to 1 mol of fluid). 



ilMfeMWtfi 







Btoatis® of the ^©at mriety of uses to whicn stibaoaic diffussrs ^y "bs 
put, tli®r® ia aTailaW® a lar^ spowat of ©xparimental data on the pepforaanc© 
and efficiency of dlTers© shapes and types of devices for traasformlng ^t% 
of the klnetiq energy of a flowing fl\ild to paresstare energy, f he most coflplet® 
eoilatloa and ejcaidnatlon of data are contained In Eefs 6 and 7> vhleh lacluds 
also a aniaber ©f references to the IJItratur©* 

fh@ dlffuser efficiency depends prliaarlly upon the rats at which the 
flow expands; for a conical diffuse r with straight ' diverging walls, the 
optlKun angle hetween the walls lies In the Interval 6 - S®, Dlffusers with 
non'-oircalar cross sections are less efficient than corresponding flow @x<* 
panders which have circles as their right sections, 

Isvertheless, hecause It Is so Important to keep the weight and drag of 
the daoted rocket power plant as low as posslhle, It may not be practical to 
us® a ssall angle of exi^nslon, sli^e that would require a dlffuser of pro<> 
hihltlvely great length; it laay he necessary, therefor®, to employ a dlffuser 
with a large angle of divergence. 

According t© Bef 7. however, "because of the free stream expansion a 
dlffuser with an angle of dlvergeace ^eater than the optiiroB «ay he used 
without the penalty of large losses, as the external diffusion acts to ''assist 
the Internal flow hy fanning out the streaallnes so that they tend to follow 
more easily the dlverglE^ walls of the dlffiiser, fiood efficiency laay there- 
for® he expected from a dlffuser which comhlnes an extei^al wl^ an internal 

llg Ik of aef 2, illustrating the variation in |f^p(the dlffuser ©fflcleney) 
as a fiaaotloa of the Ma^ number M, of the entering fluid, suwiarises tti® 



Uil^^iil 




msOlf Project ; • : , ' 



atailalJl* data on diffuser perforaaijase. Op to speeds apprtwaohlag tlia% of 
sound the diffuser efficiency is essentially constant, eqtml t© 0,85, 

Computations of Vte ducted rocket thrust, aesui^g diffuser efficiencies, 
of 100 per cent and 85 per cent, differed only 'by at most 9 per cent, farmer, 
in this report all calculatime are intends d to T@r@Bl ^xe optiffloa p@rfor»ace 
of a ducted rocket, and therefore the aatheraatically BiaxiaHiffl effieieiicy of 
the diffuse? ought to he that used in the majaerloal work, Gons@qu©ntly» ^e 
dlf fuser efficiency will he taken as 100 per cent , the smm as that of the 
free stream diffusion. 

Hence there, is no essential difference hetween the cofflpresslon as realized 
in the free stream and as aecomplished In thB ramjet diffueer; ao the diffusion 
can he Boasidered t© taksilplaeeT.eatiMiy in the atmosphisres resulting in for- 
mulae ^tch gl-re conditions at the combustion chamher entrance directly in 
terms of those prevailing in the free stream. 

Ti^ Idealisiations listed at the heginning of this Part, nov «p|J*<ld4t# 
thB total expansion of the flow, together with Bernoulli** equation for th© 
Isentropic flaw of a cot^ressihle fluid and the tow of Gonservation of fcss, 
are expressed imthematlcally he lows 

J^= t -*• ^ Me (Bernoulli* 8 equation) 

^«S-A=_L_-J (Goneervatioa of mass) 

hk^-ikf [A..u^tlon (1)] 



i-m 



^\ JA8BUiBpti©ns'<2) and (3)] . 










fhese i^lattems are easily coitfbiaed t& give a set of almdtga}.@eU8 ecfoatlons 

nrhich eaa be solved for Mj ®ad -:— in teras of the parametera Mo aact ;p i 

I 

' To l+O.ZM^ 

in these relations / has been »et equal to 1.4, followiia^ id«i^lizatioB (2). 
for givea Mq amd -^ the values of M^ and -^ can "be easily calculated tj 
an iteration process, fhen the values of -^ and ^ can he computed; wlt^ 
this knewled^ the results can be readily checked "by calculating M^ from 
the foriBula 

derived sioply fxom the Law of Conservation of Mass: 

»• Oharfcteri sties ef the Working fluid leaving the Combustion Oh^ber jLa 

Terms pt Indivi dual Pi: o perU e8..of ., the_Mtering,, Ai.T and.B^,^^ faseg, 

1- P^.. .coat^^sti-Qa chamber 

fhe ceiBbustion chsmber of the ramjet component of the ducted 
roekst is considered to be a circular cylinder, of voIxum sufficient 
to provide space for combustion, and of length vhich permits eoaplete 
secondary burning of the rocket exhaust gases before the prodcrts ©f 
the secondary reaction enter the duct nozzle. Zn addition, the 
combustion chaaber should be long enou^ to insure that the velocity 
distribution across the end section is reasonably uniform, a strictly 
constant velocity distribution being that for tAich the losses are 
a minimuBi. 



iMlHtM-AAl 





OWSGlf Project 
JPi-S4LGlf 



Such ^owtrical speelallssatloa is ^mlid beca'as® the thesTsiisal 
aimlysl@ of th& cbocted rocket is iaebspeadoBt &t ajEgr particular com- 
buation chamber ahape, except tdiat a uaifora cross sect ion is &mmmd. 
thjreugheut, requiring only that the chsmlaer be eo desigtaed that 
mixiBg and corahustioB ar9 ccmplste by ths tiss the irorking fluid 
reaches the duct nozzle. 

fha large araoxuit of momentum present in the rocket Jet« by 
providing a source of high-energy turbulent eddies, should 
aaterially assist the mixing process. 

ISieory has not been successful in predicting the length of 
chamber necessary t© secure conrplete combustion for given entramst 
conditions, propellant, and air flow. Also, because this problem 
is so new, there has not been sxxfficient time to reduce experimental 
data to design parameters and working rules. B«nce n© estimate id.ll 
be given ©f the length of chamber re<jULired to meet the objectives . 
of complete mixing and complete combustion. It M^y be possible t© 
reduce the chamber length considerably by the installation of 
Multiple rockets, ^ich would function to assist the mixing and 
burning processes. 

For example, using multiple fuel injectors, recent ramjet 
combustion clmmber tests conducted at the Jet Propulsion laboratory, 
G^^Xf, seem to place the leng^ of chamber necessary to secure 
reasonably cos^lete mixing and burning at ^0 in, independent @f the 
diameter. 







P, P, MB. Pr®40ai;lf527 ■ iW^^PPwMi-. ■ I Page 21 

OBDOIf Pro jeet ^ , 

JPL-GMieif 



It will be necessary te give careful conslderati^a to th@ eoa> 
Mstion cham'ber length, slace the external drag ia of outstanding 
importanee; for tMs reaeoa a chamber with a given cross sectional 
area should have the mlniaum pOesihle length, with due regard to 
ether faeters stush ae weigbt, siiaplioity of oonstraction, fuel 
consumption, etc. The Telocity of the air flowing through the 
central duct section is ^sall, and hence a judleieue desi^ will 
produce a ^looth, lo«>-drag, internal stjrface. 

Beference 8 presenti the theoretical perforsan^e analysis of 
a system eisploying a rocket motor in a duct, acting as an elector, 
to provide the source of power for driving propellant puarps, Sijash 
a system is similar to that of the ducted rocket, and hence the 
reference above will serve as a ^ide in th® performance analysis 
to follow, as the same basic equations (with one slight modification), 
govern the phenomena. 

for the calculations relative to the processes inside the 
coiabustion chamber, the following S^dealissations are employedi 

(a) fhe air, tne rocket ^^aust gases, and the working 

fluid are postulated to be perfect gases with constant 

specific heats. 

(S) fhe mixing of the products of the secondary burning 

is considered complete at the end of the eosbustion chamber, 

where the distribution of velocity is uniform, 
(c) frictional losses are aon-existefat , 



iAirtrn 







U) At S«etiom 2 th@ •pmamrd la m® r@ok®t jet Is 
avsWed to 1}e ^mt ©f ^e air ia the s&as plaae; the r^lmitf 
&t the jet &t this ae@tioB is defined to U t}^ e^tbaust 
-reieoitj. Since the preesure at. ^e eom^%%0n ehGMher 
entrance will Sjaorease with ^e free flight Trelecity ©f the 
tmlt, it fellows that tl^ presstire la the jet will ^mry^ 
causing the area of the reference section (^{) to change 
with Vo . However, sii^ fj^ appears always as p&r% of the 
expression fi+-^=j, where fj is at least ^oof/j it follews 
that mriatiens in the size ef f/ will not he signifiesmt. 
Hence in the numerical work -f/ will he chosen as a constant, 
interBsdiat© het^reen the ■values it aesuaes for Va = (Mo= o) 
and Vo = 1120 ft per sec ( M^ « l). 

The fandasssntal relations governing the mixing process 
are as follows 5 

(a) Condition of conservation of i^ss 

fhe mass of air and propellant ^ses entesw 
ing tiie comhustion chamber in tmit time must he 
equal to the nass of reaction products f^ss^ag 
throu^ the end section in w^% time. 

!Ehe requireiaent tlmt the mss he conserved 
is laapoted hy setting 



pz*zv^*s^^xv; = p;*^*v;, 




p. p. MB, F?©4®cr^5£7,'-/^'|i^^Sill^wOTBi'"' v. Page 23- 



%'«v, ■ ■ .or 

m, + m,' = p,"<v;. (1) 

(^) Ooadition of conseirvatlon ©f preBSTjrs f ©rees aad 

fhe difference "betm&n %h@ ®®aenta ®f tli® air 
and |>ls0pellant ga8@9 eateriag ths oomtmstlea dmmb@T, 
aad the momentum of the working fluid leasing, throu^ 
the mi. section, all in sanit tiae, aust equal the sua 
of tixe forces acting on tte flow, «hich are only the 
pressure forces o-^jr the cross section of the fluid, 
since the comibustion chamher is considered to he a 
smooth cylinder, 

fhe requirement liiat the sua of the moaeatua 
and pressure forces he conserved is iarposed hy 
setting 

Since -ft+-<V = -fa and vi=pz , 

miVi + »TiiVi'-(mj+nt^)V3" = fjCP3-Pi) (2.) 
(c) Condition of conaerration of energy 

fhe total energy of the air and rscket gases 
entering the combustion chamher, plus the heat, H , 
released during the secondary burning of the roekBt 
exhaust or additional fuel in the inf 1©w1j«' air. 



OHkl^iii) 




p. p. LAB. Projer.t ^¥,52? - ||i^|?rf^i|f|l^ "Page Zk 

OSBCIf Project 

JPL-&iaGX.Sf 



aust e(|Tial the total ener©r of the reaetioa products 
leaving %he coabustlon chamter. Because .of ths 
idealiaatioa (a) of %h.& pyevloua ssatloa {Qt p. 21) , 
the enthalpy CpT for each fl^id {air, propellaat ^ 
gas, working flnid) may 1)6 varittea as ^ -^ . <. 

fhe requirement ttet the total energy he con- 
served is iEPpoaed by setting 

S^jsaai^izing, the hasle relatioaa controlling 
the mixing and htsraiag prooesses In the comhiistton , 
ehamher ar© 



Vn^E^+m^E^+H = «.*"j5Wi')(:^-pr+i>^''') 



IVom Part 4» prop^rtl** < PijVi, p^, Wij^sfA^i) 
of th@ air entering thecbfehastionchffliher ar© knowa^ 
eiad the section ^llo;»tag will present formulae 
•showing the r«latioasl% hetwep '»' » S^ , "^-L » fn^ 

aadtl* defining t|iaa?ae*»lati«i of a rocket motor. 

foT't^tr, a\fl'»«ii, reclcet'^motor.,,ifill have a 
■ deftnittq'uanti^ of -.heat.^esier^ avi?.ilable in its 



QlDOSf Project ; 
JPL-OAMIf 







Pag« 25 



and so H is a Imoifea,^ or easily ecu^utad, wmlmr, .,_ 
Also, f 3 « ^^i-t-f^ » i^ere il is a esastaatt 

soffid«here iaterajadiate ^tt#a©a ^^^ for Mo" aaid M'o = li 

-fi is, vi^ln practical limitati@Bs, a tidiollj arM-> 

trary fundamental da sign variable, 

Finally, as pre-rlotisly reimrked, / ie tskBn as 

a mean value, given "bj 

y"= "'x^-*-''"z'l^' - 

fhere remain ®nly three tmkaoims, v^' , 9" , and 
p^' , %^ich may be obtained as the solution @f the 

set of three siiaultaneous eqtmtlons (l), (a), (3). 

tts valTiB of V3 is given as 
v" = —I— — fp, +r+ »r«iv, +miV,' ± 

\j(Pi^3 + m^v^* w,'v/f - ZCtt»r+n'i) ^ ( H+*niEi+tti(i^.)J (4) 
where 

The sigaificanoe of the (t) sign will he discussed 
later. With the value ©f Vj" given above, the ssgal** 
tudes @f <j'^ and p^" are oesiputed using (1) and {2} 
respectively s 






iVj 



Fj Tir— ; —FT- V3 , 

+J ^3 



(5) 

C6) 






m^r 




Fft^ 26 



Tha tempsFaturs T3" ®f tlie reaetlea produets at 
the d&d of th0 combustion obsmbe? ssiy lie ooi^uted 






(?) 



f Qllowiaig frsa the assumptloE that those reiaotloB 
jsrsdUiOts ebey the eg\mtloxi ®f state for perfect gases, 
Hers' R3, the equivalent s&a censtaat f©r ttie workiag 
fluid, can "tee dJtaiaed froa 



•^3 = Trv^ A "^^^ ^i ' 



In whidi n\'l aad R^ are respectively the m&@@m aad 
imdiTidual gas constants of the constituents- of ^e 
reaction products, 

lev, the addition of heat to a pis fleeing 
initially at eubsenic speed in a straight pipe serves 
to increase the temperature, pressure, and velocity; 
the speed may ^-m^ to a m&persenic imlue with tii® 
insteaitaneeus addition @f a sufficient quantity of 
heat. But if the heat is added in a continuous 
fashisn the velocity discontinuity cannot occur, aad 
the speed Id. 11 never hecome supersonic, the oazimuffi 
attainable velocity being that of seund. Since the 
fluids entering the coiabustion chamber have subsonic 
velecities and a combustion process generally evolves 



IP 



*(J.-*J„« u>.\ !'i%ss:d wJ i 




F. P. La. ■pr®^®c>^'P^27/^|||||||5^^^H|£p •■•:'' Pag© 27 



heat la a coiitiamous aannerj it Is Impossible fer th© 
worklBg fluid to attain a supersoale apssd at the 
eomhustion ehamher asit, althetji^ a sohie ir@l@city is 
not excluded. 

Formula ik) contains a (±) alga heeause it is 
th@ solutibk of a quadratic @fUatioa, hut o&lj th$ 
snsaller root, using th© (-) st0|, is of piaotieal 
sigiilf icano® . Ihls is true hocauae ySsjbn. th® H sim 
is used in (M^) th® Maeh auaher M3 = T=^==,win almya 
he less than 1, while the (-•-) sign in'Tariahlj oausss 
M3 to he greater tJmn tmity, 

fherefors the larger root ia foraareila (k) - 
corresponding to the {+) sign -, which results In th© 
uaattalnahle supersonic Telocity, mist he discarded 
as physically not significant (for the ductod rocket 
with secondary hurning proceeding in an orderly m,j) . 

finally, the conditions at the end ©f the eomm 
hustion chamher are given as single-valued funct Ions 
of properties of the entering fluids? 



^«^ P>f;-.ni.v,W< ^ Igylv; _ (10) 





p, p. UB. Project ^b'-^l tmi^lmlmmmMI ^*S® 2S 



It sbmilcL Ibe rse&Ued that 



■ e^*:^i:*^-''> E^=F7t^i^"- 



3. She yeefeat ao^er 

So far ao nvm&Tioal valusB h&re Men set into the expressioss 
(S). O), «ad (10) at)®T«. 

Howrer, fr®to the dlffuser calctdatioas. Part A, the mgaittades 
of ^ , Vj^ , Pi , p^ are knowi. 

ffix® priaed qfoaatitles, and H^ refer to the roeket jet and ©an 
he co)eaput@d ae boob as the rocket motor is defined hy the ratio of 
epecific heats (i'') of its exhaust, peepeHante, thmst (F), 
chaaher pressure (p^.), and effective exhaust velocity v(c)%,%|p^- %@ *^® 
actual exhaust Telocity. 

fhe rocket motor will not? he specified as one #iich cone^ses 
a mixture of liquid osygen and gasoline in the wei^t ratio of Jt2 
at a chamher pressure of 300 psla, for such a a© tor the effectiire 
exnauit velocity of the jet is BrtO ft per sec nhen the pressur® 
®f the air into whicii it discharges is 1^,7 psia, and the mti® ®f 
specific heats (i') of the propellaht gas is 1.285. All of thee® 
data are from Tahle H of Bef 3. fhe rocket prepellaat, mixture 
ratio, and pressure are chosen hecause of the lar^ amount ©f enero' 
that. can he released in the secondyary horning. 

Although the thrtist of the rocket ftotor will chs^ge with the 
outside pressure, such rarlatioa idll he slew, and the ftomit^^l he 
assumed constant, fhe rocket id.ll he considered to have a rated thratat 
of 500 Ih, although the final repult of the computations will :^re sent 






2, P, US, Er0j®ei SXgg;] 




Pk®-29 






• BffeetlT® extoi^^ velocity qf ^^^,4 y^^t^f^ 

from thm ymlh-k&ovn theory of the ds la-ml noazle, tlit i^rfsr- 
]]^iiee of the rocket motor can he calculated from the foraralae 

F'nmis/^ ' (11) 










a 



(12) 
(13)' 

(li^) 



Here -f^ is the area of the ri^t aectloa of the Jet n^re th® aralsieat 
pressiare is ft- 

In formula (12) / » 1.285 and Pc * 300 pela for all values #31 
p2 and vi . Siac® v^' will Ghaiaj|» slowly with p^ (for lastwaet, 
t&ea p^^lk,! psia, v/ afelji^O ft per sec, while ^d&en Pi « 27.2 psia 
C the largest value it aesuaes ia the esamplee eoupid^red in thie 
report] » '^i *5990) it appears reasoiaahle to e®a»ld»r v^ %o te 
eouetaat, equal to, say, ^kO ft per sec. With these values It te 
easy to compute §^ from (12), Since the flow aft of the rocket 
nozzle throat is sii^ersoaic, 9^ will not he affected hy ehaagee in 
tixe pressure p^^ > and heacs will he constaat. 



«ii«^»>. 




OBKJIf Project : : . 



The area of the aoszle thraat^, f^, which is aecsssarily fix®d 
far a givsn rocket motor, caa "be computed from Eq (14) for a eotk->- 
ditloB at liixich aXl of tha other goantlties ia that relation are 
kaown; for example, when Pi»14.7 psia, v( • ^0 ft pair see, 
**^* g^ 8l««8 per 8®e, /' « l»2g5. >c « 300 pal, and Sc « 0,om^i»S 
slugs por cu ft [froB Iq (12)3 ' ^^ ^^^ .^^^^ "^t * O.OO3359 sq ft. 

further, forraala (13) allows the calculation of -^ as a 
function of the outside pressure f^ , so that -^i can he computed 
with the aid of the identity 

finally, §^, the density of the rocket 'exhaust ^see where 
the pressure ii p^ and the Telocity is v/, is gi-ren ;^ 

o'-J2L_ 

by the Law of Conserratien of l^ss. 

In this manner numerical sizes of all of the priBsed variables 

appearing in the "basic formulae (S), (9)s aad. (10) are determined 

as fuBotioas of the parameters M^ and —■ and the free fll^t 

conditions p© and f^ , 

^. . Huaerieal m lue of H. the heat added by eeatpletely burning thg 
rocket exhaust in atr 

The heat energy that can be released \iy burning tl^ rocket 

exhaust pises in the combustion chamber will naturally depend upon 

the products of the reaction between the liquid ©aygea and ^sollae 

in the rocket motor, llhe composltioa of those reaction prodaot® 

is glTsa in Ifeble I, Bef 3, part of which is repeated below. 






p. p. La®. Project HX52? 
OlDOIf Project 
JPIrSiMCIf 




Be pert So. >»3 



P^ 31 



f il» I 

Prodocts of Coaibuatiea of (Saeslla® end Xii(|aid Osyr^n 

Initial Temperatare ©f faelj 35*^®f 

laitial femperature of Oxidant 1 '•£00®!' 

Mixture BatioJ Weight ratio, oxidant t© faeli 1.5a 



leaotion 
Pressure 


Oxidant 


Mixture 
latie 


Comp@siti@n - Mole fraction 


co^ 


CO 


NjO 


Hz 


300 psia 


100^ 02 


1.5 


0.033 


0.i^55 


oaai^ 


0.3SS 



:,i 



As the CO^and H^^O cannot 1^ furtdier oxidized, additional 
ener^ from the exhaust gases can only Tae ©"btained fro® th© reaction 
t>etween the CO and atmospheric ox;f^n and the H^ and atmospheric 
oxjfgea, !Ehe i^irticiilar combination of rocket propellaat, mixture 
ratio, and chamber pressure employed in the calculations was 
chosen because it was th© one with the largest percentage of Co 
and H^ in the exhaust gases. 

fhe heat energsr in the exhaust is esapited by means ®f tii® 
©bvious formula ^ 

where H is the l^at energy released in th® secondary btuming of 
^e propellaat gjases, wj^ and h^ being respectively the wei^ts 
and heats of combustion per lb of the coinponent combustible sfQ><» 
stances in the eshaust gases. 



6iyWii> 




OBI«SIf Pro j©e« Bspo^t 



Sa^ seooBd tiie rock«t aotor sheets ggg- x 32.2 * 2,5 l"b of 

matter tiirough its nozzle« 

fiien the weight of CO expelled fro® th® roeket Is 1.85 1^ per 

sec, wnlle the rocket ejects 0.1130 lb ©f H^ e-yery seeoad; these 

ntuabere are calculated in the masaer shoim belev. (!^e first BiuBber 

is the fractional part ©f a mole occt^ied loy the sabstaaee ffisatlewd 

on the same line, the second n'oatber is the molecular vei#tt« and X 

is the faetor which converts the product [joole fraetioaj •-[aoleoular 

w*i#it] to pound we l^i.) 

Wel^t of CO^ expelled from rocket « (0.033) (m) ^ « 1.^5 >^ 

Wel#t of CO expelled from rocket - (0.1f55) (28) X «, 12.73 ^ 

Weight of HiO expelled from rocket » (0.121*) (lg)x • 2.23 x 

Weight of H^ expelled frds rocket - (0.388) (2) X » JQ.776X 

17*l86x 

Since 17.19 X « 2.5, it follows that X » 0,ll+5J+. fro® this to® 

nuaerical iralues given ahove j^llow iaaediately. 

According t© Bef 9, CO evolves 1*.375 Btu per Ih tdien ce^letely 

Ijumed t® COj,^ while the coahustloa ©f H^ releases 52,050 S|^ per 

Ih. Consequently, the heat energy availahle in the rocket esdmusi is 

H « (1.85) (H.375) + (0.1130) (52.050) 

« 14,000 Btu per me 

* 5&00 Btu' per Ih ®f priaariy reaetaats. 

She coapaxative order of aagnitude ef this aaouat af energy 

will attain its proper si@&lficemoe if it is noted that the totiuL 

energy in the rocket ©aliaust, "before enteslag into the gecondear 




^DOIf Project [J :•„. ■• 

JPli-4M.MIf 



reactioa with the air la th© teet^ is appr©ximt®ly I7OO Bta p®r 11) 
of fa©l» »© Ih ©f ^solia® will yield approslmtely SO^OC^ Btu 
yi®a it is cofflpletely Inwraed, 

H®i»®s ®v0a If a fraotloa of tha l&t®ml h®at @a@r©^ ia tl» 
] eshaiast is released Ijj tb,® s®e0adar7 "buralagt th®, aM«%©d rectot 

-J ^ .. p©wr plaat should pr^dwc® a thrust tubstaatially gr®at®r ttoa 

• that ©f a aonml r®ck©t ■unit. 



0. ealetaatioR 9fPgcp9irtl®8 of the Worktag flttld at the Bad jf th® _^ftA®i 
lezala (Saetlon k) in farms of those at the lozzl® Eatmac® (§®etion 3) 

In this Part th© only imkaowj f©r whleh a Talue in soti^t is the 
velocity V4 at the exit of a aoa^le designed to escpaad the pp®BS«r© of 
th® workiag fluid to that of fee atmosphere » A givea aoazl® vill properly 
©zpand the produets of combustion for only one set of operating coaditioaso 

iTon sog th© thrust calculations are based on the asstraption tixat a 
flexible nogzl@9 which instantly adjusts itself to any givsn sitimtioa, 
ia employed aft ©f the combustion cteia^r, although It is realissed that 
sueh a d©Ylc® would miduly cosplicate an act^ml ducted rocket lastallatioa. 
Of cows®, this idi®alizati@a has its origin in the endeavor to calculate 
th® optiffliffli thrust of the rocket-ramjet duct combination. 

Analogous t© the diffuser efficieaacy, 1^0, (teflaod ia Part Ag thew 
is a so-e&ltod nozzle effieieacy factor^ i|^, descrlbsd a§ &@ fi«etl®a 
of pressure change available for accelerating the fluid | thus, 

%— -- dp - * 

Srperissatal data show that a sarsfully (tesi^ed dlsehar^ aozgle of a 
rocket motor will b® free of Aock wa^es asd eddying loases and will h&re 



■WM 







P, P» %m. Project m^^Jl, ' ^W^^^BS'HW- '- •: > ?ag® 3^ 

CfflDCIl! Prejeet ■ ■ ' -- - 



aa efficl@aey of approxiimtely 9§ ps^ c®nfe. for to® s®at part tli® tlm 
ia th® duct a®g2l@ will "b® mibBonic^ al'&ou^ the possibility of aa 
scoustle v®l©oity at the cofflibustioa cimmljer ®xi% lay arise, for tlsa lattar 
oaa» th® ©fficleacy will lie about 9S per e@at| ^. wibaonic noszl© does 
aot ©vea admit the possibility of shock wav®®^ a»d se tAi® efficisacy say 
b® ©Tea higher, fhes® coasideratioas Justify assifflptioa (4) balow, 
fh© flow thraxi# the discharge aoszle ean b® calculated ia an 
®lem®atary maaaer by th© e^loyaaat of the law of Goasertatioa of Itess 
aad the following idealizations; 

(1) The working fluid is postulated to b® a perfect gas with . 
coastaat specific heats, 

(2) All thermodyBamlc processes are consider® 4 adlaTmlio^ Th,® 
exponeal /" vhieh applies here^ caa be tatoa as a E®aa 
valu« between / and i\ with due ra^rd to the msa of 
each component entering into the secondary bumlsgi that isj 

Hove-^er, to Bioiplify coi^utations, y" %rill b® assTmsd 
constant, equal to 1.4. 

(3) Th.® nozzle expand® the working fluid tmm th® pressw® 
p3 at its entrance to the atmospheric presstw© p^ &* 
its exit without loses ?)„ * 100 per cent. 

The aathemtlcal forisalatioa of require»nt® (l)s (2), (J) 

above , 

ill 



p. P, !.«. Project md^-7 




■:t.a^ 35 



whin coHibined with a form of Berjao«lll^a equation for •^© 
isestropie flow of a cost^essible tltiid,; 

e^uivaleat to th8i>% t.$©d ia PaJ-t A, y^stslta 16 a siagl© foi^tda 
giving y^' la teras of prcrparttes of the fltdd entering tjbte 
aozzloj ■ 

(15) 






\ 



I 4 






jr" 



fhe noaisle dimeaglotts oay t)© ©ftsilf- eoaputed from tlie taw 
of CoaserTation of Mass: 






Here -fj (= -f ^ + */ ) is knorai, v^/Vj is giTeH. from tfe® 
previous result, aad 9l(f^= (Po/Pa)''^ ^^^» *^® PWfso* ^8 asd 
adiabatic flov conditions, 
lo rooapittilat©, 






With th# iralti® of y^" ealc^ated from Sq. (I5)j tAi© tfiruft of the dttcted 
rocket caa 'be computed froa ih@ formola 

fferost = cliaag® i» Bomenttm of the fluid passing througb. the duet.. 




p. P, LAS. Project MXg2?-, r-,^K^Harai«EP '■•■ > ?ag® 36 

OHBOif Project * ' 



fbe mmsntym of the working fluid leaving tEe aact ii (w, +mi') v^' ^ idxil® Ihe 
moaantum of air ©ntertag the ramjet Is m,.Va (if it la raeallsd that m^=mj). 
lano® fhruBt » {.m^*y^z)yJI -WiV^. 

2^e computed thrusts are presented in the form @f eurvaa ^ot#i3&g th® 
au^entation faoter as a ftmction of different indispendsnt variahles, ^@ 
au®seatatlon factor beixig the ratio ©f ' thrusts for the ducted and the tm- 
encldeed jet. 

The rocket eshatiet is at a hi^ ta(itperati33:« and gives 'k^ aost of its 
heat energy to the air with which it mixes, increasing the temperatiare of 
the latter, farther, it will he asstimed that the entire effect ©f hurniag 
the jet ^ses, or additional fuel sprayed into the middle section of the dactg 
is to add heat to the working fluid. :^nce it is seen that the processes t@ 
which the rocket g^ses are subjected in the coiahustioa chamher can he identi- 
fied with a certain quantity of heat added to the working fluid, 

7he ducted rocket thrust calculations are conveniently separated into 
three groups (listed helew) distinguished hy the amoimt of heat H added t@ 
the working fluid, over and above that centributed by the rocket jet in 
the simple mixing process. ' 

(i) Bo heat added^ 

for this situation it is assumed that nene of the heat 
energy stored in the combustible components of tiie exhaust is 
relaasfd; that is, the ducted rocket is considered to be 
atrletly a thrust angmanter in the usual sense, Math«aatlcally, 
the condition that n© heat (other than that present in the 
rocket jet) Is added t© the wrking fluid is expressed by 



1 aftjgiri 



,•/ 





OBDCXf Project ■ • 

JPL-fiM^Clf 



settiBg H=0 ia Iq (S) , cemputiag Vj", aad eoatiaulng th© cal- 
culatlQBB with tihs oorrespoadlng iraluss of §3 , pj , etc* 
As obser-TOd a'bOTS, for this case ^&is ducted rocket 1b esseaa^ 
tially & tiurust atigoeator; heaea tl^ results of the caXettla-* 
ti0&& indicate the perferimaee of a high-speed thrtist augmei^tor 
(Of Fig 2). 
Cil) q^gy^^te l>urBlag of the propellant gases. 

tn this izxstaace it is assiuted that the huraiag of the 
rocket jet gases ia fully realizedi that is', all of the theoret- 
ical heat of comhustion of the CO and Hj in the rocket 
eschaust is oonsidered to he released whea these suhstaaoes cobh 
hiae with the oxygen in the enterimg air. 

fh© Condition that the -propellaat gases are ooaplately 
burned in ^e air f loid.ng throu^ ^e iie'l is mathesmtioally 
expressed hy setting ,H » iO,g92,©^'^ft-W>, the heat of eoBslUBtioa 
of the Co and h^ present in the rocket exhaust,, 

figures 3 through 11 reireal tiie extent t© iitolch the thrust 
of a noraal rocket motor can he increased i&en the jet is @n~ 
closed aud completely hurned in a rsffljet das^. 

She first figure (Cf Fii^ 3) showg the ^eadeiMje of 
augaentatlon on the velocity with ^iteh the air enters the 
ramjet diffaser. Jor aigr given value of Mj the augiiWBtatien 
clearly increases with t^e free flight Maeh mugaher. With a 
fixed free fli^t velocity the au^ientatlen at first increases 
as the Mach nutther at the comhustion chamber entrance increases. 



iMi^lilS 







f. f.MB. Project HX52? ili|g«i01lMl ' ^^ 33 



811& tb.@n liegias t@ dscreas®, dxie ts a reSueed ps«sstir@ in ^@ 
xaaa^et aozKl© esatranc® at thB higher fli^t «p@@&6, 

figures k tlirou^ 7 Indicat* %hB p@rf®raaa©@ ®f a Saetsd 
rocket using a duet lar^ en&n^ %q ptirsit the sntr^ of amre 
air than that required for co^Xete secondary eo^ustion. 

The eis^loBt and most practieal dieted rooket would smplo? 
a duct with fixed dimoneioas, tten, iri.th a given diffusar 
entrane®, an increase in the fXi^t veloelty woxild cause a 
greater imes of air to floif throu^ tiae coffihustion chamher in 
a giren time. 

fhe performance results presented in figs k througih 7 ar® 
©f value ■because they dtiow cuiaolusively that wi^ a given rocket 
motor and ramjet duct the ducted rocket thfust Inereages with 
the forward velocity of the combination - or with the amount 
of air passing through the dact - primarily a® a stral#it 
augaentation effect, 

fhe ^neral result that at low flight speeds tiie thrust 
of the aeraal f#ait motor suffers a loss \it&n enclosed la a 
raffijfit diKJt (Cf tig 2) appears also in the thrust ealculatioBs 
for this case, although the decrease in ttirust is awsh sMtllsr 
than it is in (i). 

If it is found necessary ®r usefnl to eijiloy a rocket la 
the ram^t doot - as an aid to comhustion, as an instroaeat to 
facilitate ^& mrlj stages of raa^et fll#t, or as a hoostw 
for 8t:®erp®?f®T«aae« - the smll propellli^ force at Im speeds 
will he dBtriaental. 



IM^efil; 



p. p. U3, FTo^mtmm ' Wmm^HBml'mw '■ ;/ Bag® 39 

omQlf Pr®4eet : . ' : , , '.,/;; ^i>»ir^, 1«^„ '343; 



lo elittiaat® the lost tlarust at mall ▼@l@ci%i,9s, it »y 
be B^oessar^ to adiraiied to a dsal^ eo^leylng. In tlw isiti&l 
parted ©f «i« fligbt, a rocket so far aft that its a© sale pr®- 
^aets ^hiad the duct, beiag "br^ugSit ferwsrd to the mmtixmti&n 
ebamber entramce %*Lea the fll^t velocity is hi# eaem^ s® 
that an actiml awgaeatatios of thrust can "be realiaed. 
Ciii) Mgataw poseihle. a^^mt of heat a|#4i 

!l!he amowit of heat t3m% eaa he at^sorhed hy the worklag 
fluid is limited W the theeretidally derived aad experiwat- 
ally ohserved fact that the fluid velocity at the eoahuetioa 
chan'ber exit cannot exceed the local acoustic speed («&ea the 
heat applied results from a continuot^ exothermic reaction). 
For the values of M^ considered here, the latent heat of 
reaction of the Co and H^ In the exbaust Is always less than 
that required to accelerate the tiorking fluid to the leeal 
velocity of sound at the comhuetlon ehamher exit* The 
additional heat needed can he obtained hy huraing ^eeliae, 
kerosene, or a similar fuel, injected into the ©oaibustloa 
cbamher, . 

fhe peek value of H is ohtalned lay setting %he raA4^«3p>s. 
in Foraula (S) equal to zero, giving 

7'"" 

Since iSm value ®f H„^, le not ©apieyed in the later 
calculatioas, tdalle the siae of vj' is a foadaaental totaf- 



Mdiate result, it is TOre useful t® cw^jute Vj 



MAX. 



filllfi^iiitCP 



'.I 



p. PVLA®. Project 'iteSIf : " ^Pf^^^^^pKl' '"■' : ' ^ ' Bage StO 

JPL-<l&LOIf ' . "' ' ' './' ' ' " " 



'max." (mi+miHi+r"; * 
Mattxematically, th© eoaditioa that tii© aaxlmxiffi possi^X® . 

qttaatity of heat Ib addad to the werklng fltiid Is larpesed tgr 
setting 

Bl»e© this mltis of H mkes the radical la (8) ser®, oaaalag 

Vj to aasuae its maxiaa® aiz®, 
fifures 2 aad 3 haTe as their suhjeets dosted recfeets idth variahle 
diffttser aad cMtMBtiea ehsuaber areae, e@ that oaly a preaBslgtted amomt ®f 
air is allowed t© aatar th© dact (for example, ©aly so vsmh air as Is refuirad 
to support the coagilet© coabustioa of tiie rocket exhaust ^ses at 8®»@ ^vea 
fli#it talocity), aad a "mriaMe aoazle deslgaad to easpel the protosts of th® 
secoadary reactioa at atmospheric pressw© \iadsr all fli^t coadltloas. 

However, la Figs 8 throa^ 12 the dlmea8loa.8 of the ramjet dlffuser md 
©omljustloa ohaaher are supposed coastaat, the aozzle opealag aloae "beiag ©oa- 
sldered Tarlahle. Figures S through 11 show the results of a auia^rlcai' 
exattple in which the diffuser area ratio -fi/f, is allowed to asuae th® mlues 
1.5.^ 2, 3, aad M., while the coahuetioa ohMher area is fixed at k tq ft, 

fhe fxteat to wMch the aaouat of heat added to the worklag fluid affects 
the au®BSBteti®a of thrust of the aoraal rocket m%&T is repealed to figs S, 
9, 1©, aad 11. lach graph covers a rahge of free fll^t Mach auahers, 



■/;■:■ '■; -■ 



o.ii«:f%^l, f«r ftoea ea*8«j 

*f@r a coabustioa chMiher area of k eq ft, th® value of Mo at i&lch just ea©u#i 
air to support oo^lete secoadary "buralag of the exl^ust gases eaters is givea 
hy 0.0245 \/^a . &% the lower Halt of Mq , 0.12, BUffieleat air for eeaplet® 
eombustloa eater® th© duct wh0B^i/f<,<5 . 




wm 




JPL-O&Leif 



(a) lo h@at ia addsd, other, tima. tbat prsssnt la tto re@k»t J@t «b@n it 
^^&s lo ffiiz with tiae air la II10 Sueti %h@ perf^rMmee 0f tii® #^t- 
©d raete©* ia this case is represeated "by the cuty© labeled *S0 
BtJBIIIG (Hso*)". 
(%) All of the eaergy stored ia the jet gases is liberated (W htiraiag 
the CO aad H^ coss|io£^ats of the rocket eshaust) sad added to the air 
flowiag thretijg^ ttie duet ; the perf ornaaoe of the dticted roeket ia 
•^is ease is depleted 1^ the eurre laheled »GCMPLifI SICOSmHf 
BtJSIIHG 01 fSS aOCDf ISH&USS ( H « 10,892,000 ff-43)». 
(e) A quaatity of heat sufflcieat t® hriag the velocity of the'workiag 
flxild tip to that of souad is added to the mixture, this aootmt of 
heat heiag the mazimam it is possible to traaefer to tto fltzidi 
the perforaaace curve ia this case is labeled "SOTflOIBOT BIAf fO 
M4E1 M3 s 1«. 
laspefstioa of the figures uader diseussioa reveals that the ducted rocket 
thnast imj be coasiderably multiplied if fuel such as ^soliae is Imjected 
iato the eombastioa chamber at high flight speeds, tto additloaal augroatatloa 
of thrust dependiag oa the area ratio -fj/^o . fhe ©urros labeled ^SWFIOIWS 
Hmt, etc.,'' la ligs 8 through 11 reveal the exteat t© wlxieh such iaoreas® ia 
thrust follot^s froa the additioa of the fflaxiaua possible aaouat ©f heat to the 
air n owing throu^ the ramjet duct. Howver, figs S through 11 show clearly 
that large valws of H are aot alwi^e to b# desired, siaee the duet->ifet l^rmit 
whea the rocket exhauBt g^ses are completely buraed C H ' 10,892,000 ft^lb) 
is coasiderably ipfeater t^eaa that obtained with a soalc velocity at the ead 
of the combustioa chamber, evea thoupi the heat re^tuired ia the latter i^se is 
iavarlably greater thoa 10,892,000 ft-lb. 




p. P, LAB, WtqAbcV 'HX^^'t • " .' •B^^^^j^ ^ P^lMlv ' '' ' : ' : Pag® k2 

OlDQIf .Project .:', ' :.. ',,. ■ \ . ^^QT%'W<^-'y->'^ , ' ^ ' . J ; ■ 
JPL-a&LQIf ' ' • 



figure %S. @ff®rB a cosparisoB 'between ths perforsaftsce of a ducted z^cket 
and a j^iajet @f equal p}ijaloal dlmeneione constmlng fi^sX at the saae rate a@ 
tlie rocket meter, 

®xe perforaance of euch an ^equi-mlent" r^ijet nay Ise calculated in the 
elementary aanner of the ducted rocket lay recognizing that in the two pro-t 
pulsion eysteme the only essential distinction is the momenttiia ^ the entering 
fael, the exhaust gases having a momentum far in excess of that «lth niiids. 
fresh fuel is injected into tlB raajet. further, the rocket of the ducted 
rocket unit required »«( sl-ogs of propellants per sec (fuel and oxidant), 
while the ramjet needs only the fuel component (of, mass equal to m/^ ). 

That is, the three conditions of conservation of Esaas, conseriration of 
pressure forces and momentum, and conservation of energy [Of Iqs (l), (2), 
ani. (3), pp. 23 and £1+ ] , applied to the "equivalent" ramjet, "become 

*"z*K= ?;-<^"v3 (ly) 



where 






*^tUBlly, €(y- should he given as >rPt/(Vr'-i) p/r » Ho-wever^ the error committed in 
assuadng that ?iV = fx' will not influence th@ results t® an extent greater than 
the other approximations used in the analysis. ^Is follows from the fact that 
the density of the rocket exhaust jet at the heglnning of mixing, p^ , will he 
small hecause ©f the hi^ temperature of the gas's sine® any new fuel injected 
into the coB^stion chaaher enters a hody of air at a high temperature, the 
density f/^ Will also he small. The values ©f /'for tl:^ rocket @3to,U8t jet 
and the vaporized new fuel are also assumed to he the same. 



QSDOII Prefect ;, !, % 
JPL-Q4LClf 




Pag® lf3 



Wiy. a aaas ^f the f\i®l cei^aneat ©f the rocket 
pr©pellaa»t8. 
Th@ solution af the sinraltaneous eqimtions (1 ), (2_,)» (3-) fo? ^w 
unk3i©wa.s Vj" , §^ , and pj gives t^ro® foiwula® siadXar t© those torivad 
in the ducted r©ck@t calculations s 



V, = 






n\+nii' 



;j^W3-**"iVi-\ Cp,.f3*wivJ -l^^prr^ (Sy) 






(9r) 
ClOr) 



Whan the rocket prepellsmt Is specialized t® the prspellant ceabinatien ®f 
thg previous calculations, utilizing liquid gxygen-gassllne In th® rati© '}l2 
■by weight, *Ttir= oAtnl , In the ducted rocket study a ^<Xi Ih thrust af@cktt 
aj©t®r, for which mi = 0.0777 ^I'sg® P®^:' sec, was consistently U8®d, s® the 
" equivalent" raajat will har® a fuel c®nsumptlon of 0,0311 slugs of ^salia® 
per B9C, 

Since th© fuel mass flow of the equivaleht ramjet is different froa the 
pi^opellant Biass flov of the hasic rocket motor, the augmentation factor is 
in this case the ratio 



effective eafaaust veleoity 
l6, 100 ft/ see 



OIDOIS Pr© j@ot ;^ • 

JPIr»S4IjOIf 



111"! j ^ ., i'ci -^i 




V&gB kk 



One® th® physical properties of the mmjet warkiag fluid at ths neazla 
ssXtmaxio ar« lai9«m, the resalalng calei3^ati«m.3, leading t® the ooaputati@a 
of thrust and augaieatatien facter, are identical ifith th@s@ prfvieusly 
detailed. 




fiSlSl^ Preset '• 

JPt-aMiOIS 



1iN i '" i ■ j il i p If'ii' i T'n ii r' M iiiii, ^ . 

1 . ■ 

1. v-oa Karaan, ^., Malisa^ f. J., S^2iB3&ex>fl9ld» M., and fsies, E« @., 
, »C©i^>aj?«tiTB St«dy of J©t Propalaisa Sy»1>«ffl8 as Applied t© Mleslles 

i aad frasasitle Aivejr&tt^t MmT&&^xm JHi-2, <Jet Pr@palslda la.'b^ratorj* 

SAI^If , Mardi 2S, lfi4. 

■5 . ■ ■ ■ ' 

1 2. Taagrp®^ B. f, , "Ssfclmted fiaajet P«rf®£waiie©*, Prosress SepQfl So. 3-fl, 

J©% Pr0p^8i®a laberatery, (RLSIf, lowalwr 20, 19^^, 

( 3, "Jet Sjeetors and 4a©wa*ati®a.* Bepsrt a-Jlt, Halted Aircraft Oorp©ratl@a, 

ae8©ai?eli MTl0l«a» Oetalwr 2Jt, igifl, . , 

l*. ¥©is8l»lutih, M., ^Pepferoaac© ®f Jet Motors Based on. fhesrstleal 0aleala« 
\ tioas with Speeial Bsfereaee t« the MQttld Oxygen-^solla& ^^f®i»Wt s 

Cembiaatloa" , Progress H©p©rt H®. l-l6» Jet Prspalsisa La1)srat9ry, SAldlfj 
, April 2S, 19M*. 

^ 5. Mlllikaa, 0. B., " A© r® dynamics of the Alrplaae." Jmx Wiley aad Seas, 

lew f®rk, 19IH (p.l). 

6. Pattersoa, G. I., *Mod©ra Dtffuser Itealga." Aircraft Saglaeeriag, ?©1. 10, 
Septeaher. 1938 CpPv 267-273). 

7, Patt©raoa» G. I., ^fflaa Desiga. ©f A®roplaae Ducts." Aircraft laglaeeriae, 
Tel. U, July, 1939 (pp. 263-268). 

> 8, Toa Imrnn^ Tti.t fslea, S. S., aad Oaarl^t, S. B., ^A Stiadyff /^ii® 

P®88ibillty ©f Ustag to© Ej©et®r Actiea of the Jet a© a S©arsa ©f P®%f®r 
for Driviag Propellaat Pasap8«j Pregresa Bepert So. I-7, Jet Proptaslea 
Laljoratory, &ALOIf, Jaly 27. I9l*3» 

9. Bandbeok of Qiemistry aad i^sl<^8. Chemical Ei3h1»er Publishiag Co., 
Clerelaad, 28^ Sditiea (p. 11^4-6).. 

10. Mel©t> I. f., «0a Jet Proptilsloa,^ Scieatific Apericaa, April, I926 
(pp. 2fo6-268). 

s ' . , ■ ■ ■ 

11, Jaeel}©, B. «uid Shsemfcer, J. M., "Tests ©a anast Aogaeators f#r 
Jet Pr©pul8l®a.« HAOA fechatcal Hot© So. H31. September, 1932. 




R R LAB PROJECT MX CS7 
O80CI3" fROJECT . 

JPI,«»6ALCIT 






PAGE 46 

Fie. I 







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i 



I 



i 

I 



I 




p. p. LAB. PROJECT MX5Z7 
ORDCIT PROJECT 
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PA6E ^> 
FIG. 2 




p.p. LAB PROJECT MX 527 
OROCIT PROJECT 
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PACE 4CS 
FIG. 9 




RP. LAB PROJECT MX527 
ORDCIT PROJECT 
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PAGE 49:S» 
FI6. 4 




t>P. LAB PROJECT MX 527 
ORDCIT PROJECT 
JPL-6ALCIT 



PA6E SO^^o 
FI6.5- 




RP.LAB PROJECT MX527 
ORDCIT PROJECT 




eP.LAB PROJECT MX527 
ORDCrr PROJECT 
JPL-GALCIT 



W6E 5a(E, 

FIG. 7 




p. p. LAB PROJECT MX 527 
OROCIT PROJECT 
JPL-6ALCIT 



PAGE 54S» 
FIS. 9 




p.p. LAB PROJECT MX527 
OROCIT r PROJECT 



PAGE SS-*- 







RP. LAB PROJECT MX527 
OROCIT PROJECT 
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ftp. LAB PROJECT MX527 

oRocrr PROJECt 



RftSE 5T«iV 






i ili iJft#lA''^ #-ffil 


P, P, MB. Projfec^saSc:? 


ii:i»l?i.!iiii 


OBDeiT Project 
JPL-Q4LCI3? 


■*'mwvfid':;>3 



los. X throu^ 3» lacluslre,' to Cosmandlag General, Air fecSmioal Serric® 
Comffland, Wrtght field, Dayton, Ohio (Atteatleas Poww Plaafe laljoratory. 
Jet Projmlsloa Branch) j 

Hos. 4 throu^ 31* itiolufive, to Colonel B. S. Mesick, Chief, Besearch and 
Developsssnt Service Sub-Office (Bsci^t), Office of the Chief of Ordnance, 
California Institute of Technology? 

So8. 32 and 33 to Colonel E. H. iddy. Air fechnical Service Command, 
Engineering Liaison Office, California Institute of Technology; 

No, 34 to Arifly Air forces, Materiel and S®rvices, The Penta©>n, Washington 25, 
D. C. (Attention J Lieutenant Colonel C. D. Gasser) | 

lo. 35 to Chief, San francisco Ordnance District (Attention? Artillery 
ArauiQnit ion Branch); 

loa. 3fa and 37 to JPL File, California Institute ©f Tschaoloar; 

lo. 32 to Jet Propulsion Course Files, California Institute of fechaoloari 

So. 39 *o i>r. Theodore von Karman, Eoom 4D, 0128, Pentagon Building, 
Washington 25, D. C.j 

lo. kO t® Dr. C, B. Millikan, California Institute of feehnology; 

lo. 41 to Dr. f. J. Malina, California Institute of Technolo^; 

So. k2 to Prof. H, C. Hottel, Massachusetts, institute of Technolo^j 
Cafflhridge, Massachusetts; 

So. k3 to Officer in Charge, BtiAer, Project TED Ho. MS 3IJ.OI, U. S. laval 
Engineering Experiment Station, Annapolis, Maryland; 

lo. 44 to Chief, Btireau of Aeronautic s, Jfevy Department, Washington 25, D. C. 
CAttentloni Ships* Installation Bz«.nch) ,